1. Field of Invention
This invention relates generally to fluid flow measurement, and more particularly to a method and systems based thereon for conducting a multiple velocity traverse of a flow stream cross section having a known shape and area in order to determine fluid flow therethrough.
2. Measurement Problem
The concern of the present invention is with the measurement of flow rate through a stream cross section of known but not restricted shape and having a known area A. The product of area A by a sample point velocity is generally unsatisfactory because of wide departures of local velocities from the area average. Traversing the area A with a movable point velocity sensor or with a line of many fixed point velocity sensors so as to obtain a single line average velocity u on a single traversing chord affords only a partial improvement. To further improve measurement accuracy, it is known to use a number n of traverse chords to yield line average velocities u.sub.i modified by weight factors w.sub.i to reduce integrated flow uncertainty as n is increased.
Many methods of conducting multiple velocity traverses on a number n of chords traversing a flow stream cross section are available, some suitable in rivers or open channels presenting many stream cross section shapes, and some in circular pipes. Some are more suitable in gas flows at elevated Mach number, and others are more suitable in liquid or slurry flows.
The state of the art of multiple velocity traversing is such that any one of n traverse chord average velocities u.sub.i can often be determined with an uncertainty of a few hundredths of a percent, whereas integrated flowrate measurements of commensurate accuracy have been very costly. Even in case of the idealized analytical velocity profiles approximated in straight circular pipes and called axisymmetric, such as the curves of velocity versus radius shown in FIG. 1, the problem of employing an economically acceptable number n of traverses and at the same time maintaining a single calibration constant of acceptable accuracy has presented considerable difficulty.
In FIG. 1, curve 1 is the flat profile found occasionally in fibrous slurry flows or in the flows of gases and liquids for short distances after a faired contraction. Curve 2 is a representative turbulent profile, while curve 3 is the familiar parabolic profile of laminar flow found at low velocity or high viscosity. Curves 2 and 3 are found at pipe cross sections preceded by long straight lengths of pipe of uniform area and circular section. Curve 4 is the conical profile, convenient as a severe approximation of the peaked velocity profiles found downstream of gradual and sudden enlargements. The actual profiles are difficult to predict, but these examples permit some evaluation of the calibration stability to be expected of multi-traverse flow measurements which use various chord configurations and weight factors.
For specified accuracy, the required number n of traverses may be minimized if traverse line choices and weight factors are in accordance with the abscissas x.sub.i and weight factors w.sub.i of one of the well-known approximations known as numerical integration methods. These are found in mathematical references such as the U.S. Department of Commerce publication, National Bureau of Standards Applied Mathematics Series 55, "Handbook of Mathematical Functions" (hereinafter referred to as NBS Math series 55).
3. Prior Art
Malone et al. U.S. Pat. No. 3,564,912 discloses a multi-traverse measurement employing traverse chord locations which are appropriate to flow streams of unknown shape and area. A single arbitrary chord of width M is measured and bisected. The traverse chords having lengths L.sub.i, all normal to the width chord M, are placed at fractional lengths m.sub.i from the bisector of M. Values x.sub.i are taken to be ratios of the fractional lengths m.sub.i to the semi-widths .+-.M/2 so that x.sub.i =2m.sub.i /M. Means are provided for measuring the product u.sub.i L.sub.i for each of n traverse chords, modified by certain weight factors w.sub.i appropriate to the x.sub.i previously chosen and for computing an estimated volumetric flow rate Q from the following equation: ##EQU1##
Malone et al. choose values of x.sub.i and w.sub.i to correspond to the abscissas x.sub.i and weight factors w.sub.i provided by one of the well-known numerical integration methods, such as the Gaussian integration method, or alternately the Chebyshev or the Lobatto integration method, and so approximate the integration of the differential form f(x)dx, where f(x) is the product uL. When, however, this method is applied not to a stream cross section of unknown shape but rather to a shape which is known; that is, where L is a known function of x, the system accuracy suffers from the method which makes no provision for taking advantage of this knowledge.
For example, a system of Malone et al. employing n=3 chords traversing a circular pipe with Gaussian integration will experience calibration factors which vary by 5% over the extremes of axisymmetric profiles shown in FIG. 1. Attempts to use this three-traverse system in a long straight pipe in the transition range of Reynolds numbers of 2000 to about 4,000 where there are abrupt step changes of the velocity profile between curves 2 and 3, FIG. 1, will produce oscillations of the calibration factor of nearly 2%. In this application, the system of Malone et al. would require n=6 traverse chords to approach in flow measurement uncertainty the uncertainty of a few hundredths of one percent with which the individual traverse values u.sub.i can be measured. Moreover, this system requires that traverse chords must be parallel, however inconvenient such an array of traverses may be.
The Wyler U.S. Pat. No. 3,940,985 discloses a method which partially reduces these deficiencies for circular pipes only. For such a pipe of radius R it is known that L/R=2.sqroot.1-(m/R).sup.2 for which Wyler avails himself of a numerical method for approximating the integration of the differential form f(x).sqroot.1-x.sup.2 dx, found in Section 25.4.40 on page 889 of the NBS Math Series 55. When Wyler employs n=3 traverse chords, calibration factors will vary by more than 3% when velocity profiles of FIG. 1 are encountered, while step changes and fluctuations of the calibration constant of about two-thirds of one percent will be found in the transition between laminar and turbulent flow in a long straight pipe. Moreover, the Wyler method is not applicable if the flow cross section is other than circular, and requires that the traverse chords be parallel.
Additional multi-traverse flow measurement systems include those disclosed in the Baker et al. U.S. Pat. No. 4,078,428, the Brown U.S. Pat. No. 4,102,186 and the Lynnworth U.S. Pat. No. 4,103,551. All these systems are limited to circular pipes. Moreover, all give equal weight to the various traverse chords. All place the n chords essentially at a mid-radius position, which is to say, tangent to a circle whose radius m is a specified fraction close to one-half of the pipe radius R, ranging from a low of 0.497R for one of Lynnworth's arrangements to a maximum of 0.6R for one of Brown's.
Thus, in all these cases, the advantage of using more than one traverse chord is limited to a partial tendency to average out the variations that a single mid-radius traverse chord would encounter in the event of some slight asymmetry of a relatively flat velocity profile such as the turbulent profile curve 2 of FIG. 1. When encountering the various axisymmetric profiles of FIG. 1, these essentially mid-radius chord position systems have calibration constants which are those of a single traverse using an off-center chord. The exact choice of chord position makes it possible to equalize the calibration constants for any two normal profiles expected in long pipes, but any such position is exceptionally vulnerable to a sharp profile, with typically an eight percent shift from a normal profile to the cone profile curve 4 of FIG. 1. Moreover, none of these systems is applicable to non-circular flow stream cross sections.
The above-identified prior art patents deal with ultrasonic or acoustic methods for effecting flow measurement. However, total flow may also be determined from point measurements of velocity by means of pitot tubes of the type commonly employed for measuring linear velocity (see Handbook of Fluid Dynamics--McGraw Hill, 1961, pp. 14-7 through 14-9), hot wire anemometers as well as propeller meters attached to the ends of a probe which allow positioning of the sensor along a chord within the flow stream. The use of ultrasonics to obtain the chord average velocity yields the value u directly; whereas in point sampling methods, u has to be calculated.